Problem 61 - College Algebra

Question

What is the complex conjugate of \(2+3 i ?\) What happens when you multiply this complex number by its complex conjugate?

Step-by-Step Solution

Answer

The complex conjugate of \(2+3i\) is \(2-3i\). When you multiply this complex number by its complex conjugate, the result is \(-5\).

Step 1: Find the Complex Conjugate

The complex conjugate of a complex number is obtained by flipping the sign of its imaginary part. The given complex number is \(2+3 i\), so the complex conjugate will be \(2-3 i\).

Step 2: Multiply the Complex Number by its Conjugate

To multiply two complex numbers, distribute the product across each term following the rule \(i^2=-1\). Therefore, \((2 + 3i) * (2 - 3i) = 2*2 + 2*-3i + 3i*2 -3i*-3i = 4 - 6i + 6i -9 = 4 - 9 = -5\).

Step 3: Final Answer

The complex number \(2 + 3i\) multiplied by its complex conjugate \(2 - 3i\) equals to \(-5\). The result isn't a complex number because the imaginary parts cancelled out. This is a property of a complex number multiplied by its conjugate: it always results a real number.